Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

cc-Shapley: Measuring Multivariate Feature Importance Needs Causal Context

Jörg Martin, Stefan Haufe

Abstract

Explainable artificial intelligence promises to yield insights into relevant features, thereby enabling humans to examine and scrutinize machine learning models or even facilitating scientific discovery. Considering the widespread technique of Shapley values, we find that purely data-driven operationalization of multivariate feature importance is unsuitable for such purposes. Even for simple problems with two features, spurious associations due to collider bias and suppression arise from considering one feature only in the observational context of the other, which can lead to misinterpretations. Causal knowledge about the data-generating process is required to identify and correct such misleading feature attributions. We propose cc-Shapley (causal context Shapley), an interventional modification of conventional observational Shapley values leveraging knowledge of the data's causal structure, thereby analyzing the relevance of a feature in the causal context of the remaining features. We show theoretically that this eradicates spurious association induced by collider bias. We compare the behavior of Shapley and cc-Shapley values on various, synthetic, and real-world datasets. We observe nullification or reversal of associations compared to univariate feature importance when moving from observational to cc-Shapley.

cc-Shapley: Measuring Multivariate Feature Importance Needs Causal Context

Abstract

Explainable artificial intelligence promises to yield insights into relevant features, thereby enabling humans to examine and scrutinize machine learning models or even facilitating scientific discovery. Considering the widespread technique of Shapley values, we find that purely data-driven operationalization of multivariate feature importance is unsuitable for such purposes. Even for simple problems with two features, spurious associations due to collider bias and suppression arise from considering one feature only in the observational context of the other, which can lead to misinterpretations. Causal knowledge about the data-generating process is required to identify and correct such misleading feature attributions. We propose cc-Shapley (causal context Shapley), an interventional modification of conventional observational Shapley values leveraging knowledge of the data's causal structure, thereby analyzing the relevance of a feature in the causal context of the remaining features. We show theoretically that this eradicates spurious association induced by collider bias. We compare the behavior of Shapley and cc-Shapley values on various, synthetic, and real-world datasets. We observe nullification or reversal of associations compared to univariate feature importance when moving from observational to cc-Shapley.
Paper Structure (40 sections, 7 theorems, 33 equations, 11 figures, 1 table, 2 algorithms)

This paper contains 40 sections, 7 theorems, 33 equations, 11 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Comparison with the literature.
  3. Background
  4. Basics of Causal Inference
  5. Collider bias
  6. Suppressor Variables and Collider Bias.
  7. Univariate importance is insufficient.
  8. Methodology
  9. Using the causal context
  10. A causal version of Shapley values.
  11. Why not use $\mathbb{E}[Y|\mathrm{do}(X_j,\mathcal{S})]$ instead?
  12. Estimation
  13. Simple cases.
  14. Backdoor adjustment.
  15. Using the SCM.
  16. ...and 25 more sections

Key Result

Proposition 3.2

Given a feature $X_j$ with $X_j \perp\!\!\!\perp_{\mathcal{G}} Y$ we have $I_{\mathrm{do}(\mathcal{S})}(X_j)=0$ for any $\mathcal{S}\subseteq \mathcal{F}\backslash\{X_j\}$. This implies that we have $X_j \perp\!\!\!\perp_{\mathcal{G}} Y \Rightarrow \phi_{cc}(X_j)=0$ for $\phi_{cc}$ as in eq:shapley_

Figures (11)

  • Figure 1: Causal graph for Example \ref{['ex:diabetes_breakfast']} (top), together with the according results for conventional Shapley values \ref{['eq:shapley_values']} (bottom - left) and the cc-Shapley values \ref{['eq:shapley_values_do']} (bottom - right). Experimental details are in Appendix \ref{['app:details_on_diabetes_breakfast']}.
  • Figure 2: Comparison of the regression coefficients $b_{X_1|X_2}$ (left) and $b_{X_1|\mathrm{do}(X_2)}$ (right) with $b_{X_1}$ for randomly sampled linear SCMs with 9 variables. The color encodes the extent to which $X_2$ acts as collider, cf. Appendix \ref{['app:details_on_linear_SCM_experiment']}.
  • Figure 3: Shapley (left) and cc-Shapley (right) values for the nonlinear diabetes example considered in Section \ref{['sec:experimental_results']}.
  • Figure 4: Various terms arising in the computation of the conventional observational Shapley values \ref{['eq:shapley_values']} and the cc-Shapley values \ref{['eq:shapley_values_do']} for the (more complex) diabetes example introduced in Section \ref{['sec:experimental_results']}. The upper row shows (with standard error) the context-free univariate term that coincides for \ref{['eq:shapley_values']} and \ref{['eq:shapley_values_do']}.
  • Figure 5: Causal graphs for two datasets used in Section \ref{['sec:experimental_results']}. For the diabetes example (left) the meaning of the variables are: blood glucose $G$, average sugar $H$, BMI $B$ and presence of type 2 diabetes $Y$ (target). For the example from sachs2005causal (right) the variables are various proteins with known causal graph. The protein PKA is here chosen as target.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Example 1.1: Breakfast and diabetes
  • Example 2.1: Univariate importance is not enough
  • Definition 3.1
  • Proposition 3.2: SAP for Definition \ref{['def:importance_do']}
  • proof
  • Lemma 3.3: Irrelevant context
  • Lemma 3.4: Intervention equals observation
  • Lemma B.1: Conditional expectation as optimally trained model
  • Remark B.2
  • proof
  • ...and 6 more